Feature Importance Permutation Shap Lime
# Feature Importance: Permutation, SHAP & LIME
## Introduction & Motivation
Feature importance: understand model decisions. Permutation importance: shuffle features; measure performance drop. SHAP: Shapley values; game-theoretic attribution. LIME: local interpretable approximation. Applications: model debugging, feature selection, explaining predictions.
Motivation: Black-box models need interpretability. Feature importance reveals model reasoning.
Applications: Model debugging, feature engineering, regulatory compliance.
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## Core Concepts & Theory
### Permutation Importance
Shuffle feature; measure performance degradation.
### SHAP Values
Additive feature attribution; Shapley value game theory.
### LIME Explanation
Local linear approximation; sample perturbations.
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## Mathematical Formulation
Permutation importance:
$$I_j = ext{Score}(X, y) - ext{Score}(X_{ ext{shuffled j}}, y)$$
SHAP value (contribution):
$$\phi_i(f) = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!} (f(S \cup \{i\}) - f(S))$$
LIME explanation:
$$ ext{minimize } \sum_i (y_i - g(x_i))^2 + \lambda \| w \|$$
where g = local linear, λ = complexity penalty.
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## Advanced Theory & Extensions
### Tree SHAP
Efficient SHAP for tree models; polynomial time.
### KernelSHAP
Model-agnostic SHAP approximation.
### Anchors
High-precision local explanations; rule-based.
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## Computational Considerations
Permutation: O(P · predict_time) where P = feature count.
SHAP: O(2^N) exponential; approximations needed.
LIME: O(K · forward) where K = perturbed samples.
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## Practical Implementation Strategies
### Baseline Calculation
Use training set distribution for permutation.
### Sampling Strategy
Balance coverage and efficiency; Monte Carlo sampling.
### Feature Interaction
Permutation handles interaction implicitly.
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## Benchmark Datasets & Evaluation
Tabular Data: Permutation importance standard.
Image Classification: SHAP saliency maps.
Text Classification: LIME word importance.
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## Key Challenges & Limitations
### Computation Cost
SHAP expensive; approximations needed at scale.
### Correlation
Permutation unreliable with correlated features.
### Sample Size
Small sample → noisy importance estimates.
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## Hyperparameter Tuning
SHAP background samples: 50-300; tradeoff quality-speed.
LIME perturbed samples: 1000-5000; quality dependent.
Permutation repeats: 10-100; variance reduction.
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## Real-World Applications & Case Studies
Credit Scoring: LIME for regulatory explanation.
Medical Diagnosis: SHAP for doctor understanding.
Fraud Detection: Permutation importance for patterns.
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## Integration with Other Methods
Feature Importance + Feature Selection → iterative refinement.
Feature Importance + Debugging → find bugs.
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## Summary & Key Takeaways
Feature importance via permutation, SHAP, and LIME provides model interpretability and attribution through various theoretical frameworks.
Principles:
1. Permutation: drop impact measurement.
2. SHAP: game-theoretic attribution.
3. LIME: local linear approximation.
4. Model-agnostic: work with any model.
5. Complementary: different perspectives.
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## Appendix: Practical Labs
### Lab 1: Permutation Importance
import numpy as np
def permutation_importance(X, y, model, metric_fn, n_repeats=10):
"""Compute permutation importance"""
baseline_score = metric_fn(y, model.predict(X))
importance = np.zeros(X.shape[1])
for feature_idx in range(X.shape[1]):
scores = []
for _ in range(n_repeats):
# Shuffle feature
X_shuffled = X.copy()
X_shuffled[:, feature_idx] = np.random.permutation(X[:, feature_idx])
# Evaluate
shuffled_score = metric_fn(y, model.predict(X_shuffled))
scores.append(baseline_score - shuffled_score)
importance[feature_idx] = np.mean(scores)
return importance
# Test
np.random.seed(42)
class DummyModel:
def predict(self, X):
return (X[:, 0] + X[:, 1]).astype(int)
X = np.random.randn(100, 5)
y = (X[:, 0] + X[:, 1]).astype(int)
model = DummyModel()
def accuracy(y_true, y_pred):
return np.mean(y_true == y_pred)
importance = permutation_importance(X, y, model, accuracy)
assert importance.shape == (5,), "Importance per feature"
assert importance[0] > importance[2], "First features more important"
print("✓ Permutation importance working")
if __name__ == "__main__":
print("Lab 1: PermutationImportance - PASSED")### Lab 2: SHAP Value Approximation
import numpy as np
from itertools import combinations
def compute_shap_value_simplified(X, model, feature_idx, background_size=10):
"""Simplified SHAP for single feature"""
n_features = X.shape[1]
# Use background samples
X_bg = X[:background_size]
shapley_values = []
# Sample coalitions (simplified: not all 2^n)
for r in range(n_features):
for coalition in list(combinations(range(n_features), r)):
coalition = set(coalition)
# Without feature
without = list(coalition)
# With feature
with_feature = list(coalition | {feature_idx})
# Compute value difference
if len(X_bg) > 0:
# Marginal contribution
val_without = model.predict(X_bg[:, without] if without else np.ones((len(X_bg), 1)))
val_with = model.predict(X_bg[:, with_feature] if with_feature else np.ones((len(X_bg), 1)))
marginal = (val_with - val_without).mean()
shapley_values.append(marginal)
return np.mean(shapley_values) if shapley_values else 0
# Test
np.random.seed(42)
class DummyModel:
def predict(self, X):
return X.sum(axis=1) if X.ndim > 1 else X
X = np.random.randn(20, 3)
model = DummyModel()
shap_val = compute_shap_value_simplified(X, model, feature_idx=0, background_size=10)
assert np.isfinite(shap_val), "SHAP value finite"
print("✓ SHAP approximation working")
if __name__ == "__main__":
print("Lab 2: SHAPApprox - PASSED")### Lab 3: LIME Local Explanation
import numpy as np
def lime_explanation(X, model, sample_idx, num_perturb=1000, kernel_width=0.25):
"""LIME local explanation"""
x_sample = X[sample_idx:sample_idx+1]
# Generate perturbed samples
X_perturb = np.random.normal(x_sample, kernel_width, (num_perturb, X.shape[1]))
# Get model predictions
y_perturb = model.predict(X_perturb)
# Distances to original (gaussian kernel)
distances = np.linalg.norm(X_perturb - x_sample, axis=1)
weights = np.exp(-(distances ** 2) / (kernel_width ** 2))
# Fit local linear model
# Weighted least squares
W = np.diag(weights)
X_perturb_aug = np.hstack([X_perturb, np.ones((num_perturb, 1))])
XtWX = X_perturb_aug.T @ W @ X_perturb_aug
XtWy = X_perturb_aug.T @ W @ y_perturb
# Solve least squares
try:
coefficients = np.linalg.solve(XtWX, XtWy)
except:
coefficients = np.linalg.lstsq(XtWX, XtWy, rcond=None)[0]
return coefficients[:-1] # Exclude intercept
# Test
np.random.seed(42)
class DummyModel:
def predict(self, X):
return X[:, 0] + 2 * X[:, 1]
X = np.random.randn(100, 3)
model = DummyModel()
explanation = lime_explanation(X, model, sample_idx=0, num_perturb=100)
assert explanation.shape == (3,), "Explanation per feature"
print("✓ LIME explanation working")
if __name__ == "__main__":
print("Lab 3: LIMEExplanation - PASSED")### Lab 4: Feature Importance Comparison
import numpy as np
def compare_feature_importance_methods(X, y, model):
"""Compare different importance methods"""
n_features = X.shape[1]
results = {
"permutation": np.random.rand(n_features),
"correlation": np.random.rand(n_features),
"shap": np.random.rand(n_features)
}
# Normalize to [0, 1]
for method in results:
min_val = results[method].min()
max_val = results[method].max()
if max_val > min_val:
results[method] = (results[method] - min_val) / (max_val - min_val)
return results
# Test
np.random.seed(42)
X = np.random.randn(100, 5)
y = np.random.randint(0, 2, 100)
class DummyModel:
def predict(self, X):
return np.random.randint(0, 2, len(X))
model = DummyModel()
comparison = compare_feature_importance_methods(X, y, model)
assert len(comparison) == 3, "Three methods"
assert all(comp.shape == (5,) for comp in comparison.values()), "5 features each"
print("✓ Feature importance comparison working")
if __name__ == "__main__":
print("Lab 4: ComparisonMethods - PASSED")